DOCSLIB.ORG

Instructions to Prepare a Paper for the European Congress on Computational Methods in Applied Sciences and Engineering

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Instructions to Prepare a Paper for the European Congress on Computational Methods in Applied Sciences and Engineering VIII International Conference on Textile Composites and Inflatable Structures STRUCTURAL MEMBRANES 2017 K.-U.Bletzinger, E. Oñate and B. Kröplin (Eds) DESIGN AND CONSTRUCTION OF THE ASYMPTOTIC PAVILION Eike Schling*, Denis Hitrec†, Jonas Schikore* and Rainer Barthel* * Chair of Structural Design, Faculty of Architecture, Technical University of Munich Arcisstr. 21, 80333 Munich, Germany e-mail: [email protected], web page: http://www.lt.ar.tum.de † Faculty of Architecture, University of Ljubljana Zoisova cesta 12, SI – 1000 Ljubljana, Slovenia e-mail: [email protected] - web page: http://www.fa.uni-lj.si Key words: Asymptotic curves, Minimal Surfaces, Strained Gridshell. Summary. Digital tools have made it easy to design freeform surfaces and structures. The challenges arise later in respect to planning and construction. Their realization often results in the fabrication of many unique and geometrically-complex building parts. Current research at the Chair of Structural Design investigates curve networks with repetitive geometric parameters in order to find new, fabrication-aware design methods. In this paper, we present a method to design doubly-curved grid structures with exclusively orthogonal joints from flat and straight strips. The strips are oriented upright on the underlying surface, hence normal loads can be transferred via bending around their strong axis. This is made possible by using asymptotic curve networks on minimal surfaces 1, 2. This new construction method was tested in several prototypes from timber and steel. Our goal is to build a large- scale (9x12m) research pavilion as an exhibition and gathering space for the Structural Membranes Conference in Munich. In this paper, we present the geometric fundamentals, the design and modelling process, fabrication and assembly, as well as the structural analysis based on the Finite Element Method of this research pavilion. Figure 1 Prototype of an asymptotic gridshell. The structure is built from straight strips of steel. The lamellas are oriented normal to the design surface. All slot joints are identical and orthogonal. Image: (Eike Schling) Eike Schling, Denis Hitrec, Jonas Schikore and Rainer Barthel Figure 2: Grid structure based on asymptotic curves: The model is built from straight strips of beech veneer. All joints are orthogonal. Image: (Denis Hitrec) 1 INTRODUCTION There are a number of design strategies aiming to simplify the fabrication and construction process of doubly-curved grid structures. Therein, we can distinguish between discrete and smooth segmentations 3. One strategy, to build smoothly curved structures relies on the elastic deformation of its building components in order to achieve a desired curvilinear geometry from straight or flat elements 4. Consequently, there is a strong interest in the modelling and segmentation of geometry that can be unrolled into a flat state, such as developable surfaces 5. Recent publications have given a valuable overview on three specific curve types – geodesic curves, principal curvature lines, and asymptotic curves (Fig. 3) – that show great potential to be modelled as developable strips 6. Both geodesic curves and principle curvature lines have been successfully used for this purpose in architectural projects 7. However, there have been no applications of asymptotic curves for load-bearing structures. This is astounding, as asymptotic curves are the only type which are able to combine the benefits of straight unrolling and orthogonal nodes (Fig.2).1, 2 In this paper we present a method to design strained grid structures along asymptotic curves on minimal surfaces to benefit from a high degree of simplification in fabrication and construction. They can be constructed from straight strips orientated normal to the underlying surface. This allows for an elastic assembly via their weak axis, and a local transfer of normal loads via their strong axis. Furthermore, the strips form a doubly-curved network, enabling a global load transfer as a shell structure. 2 In Section 2, we describe the geometric theory of curvature and curve networks. In Section 3 we introduce our computational design method of modelling minimal surfaces, asymptotic curves and networks. In Section 4, we implement this method in the design of a research pavilion for the Structural Membranes Conference. In Section 5, we discuss the fabrication, construction details and assembly by means of two prototypes, in timber and steel. Section 6, gives insights into the local and global load-bearing behavior, and describes the structural analysis based on Finite Element Method. We summarize our results in Section 7 and conclude in Section 8, by highlighting challenges of this method, and suggesting future investigations on structural simulation and façade development. 2 Eike Schling, Denis Hitrec, Jonas Schikore and Rainer Barthel 2 FUNDAMETALS 2.1 Curvature of curves on surfaces To measure the curvature of a curve on a surface, we can combine the information of direction (native to the curve) and orientation (native to the surface) to generate a coordinate system called the Darboux frame (Fig.3 right). This frame consists of the normal vector z, the tangent vector x and their cross-product, the tangent-normal vector y. When moving the Darboux frame along the surface-curve, the velocity of rotation around all three axes can be measured. These three curvature types are called the geodesic curvature kg (around z), the 6 geodesic torsion tg (around x), and the normal curvature kn (around y) . z normal vector y kg kn r1 osculating circle darboux frame tg r x r2 tangent vector curvature k = 1 / r Gaussian curvature K = k1 x k2 kn = normal curvature mean curvature H = ( k1 + k2 ) / 2 kg = geodesic curvature tg = geodesic torsion Figure 3: Definitions of curvature. Left: Curvature of a curve is measured through the osculating circle. Middle: The Gaussian or mean curvature of a surface is calculated with the principle curvatures k1 and k2. Right: A curve on a surface displays normal curvature kn. geodesic curvature kg and geodesic torsion tg. 2.2 Curvature related networks Certain paths on a surface may avoid one of these three curvatures (Fig. 4). These specific curves hold great potential to simplify the fabrication and construction of curved grid structures. Geodesic curves have a vanishing geodesic curvature. They follow the shortest path between two points on a surface. They can be constructed from straight, planar strips tangential to the surface. Principle curvature lines have a vanishing geodesic torsion — there is no twisting of the respective structural element. They can be fabricated from curved, planar strips, and bent only around their weak axis. Their two families intersect at 90 degrees. Asymptotic curves have a vanishing normal curvature, and thus only exist on anticlastic surface-regions. Asymptotic curves combine several geometric benefits: They can be formed from straight, planar strips perpendicular to the surface. On minimal surfaces, their two families intersect at 90 degrees and bisect principle curvature lines. 1 geodesic curves principle curvature lines asymptotic curves Figure 4: Surface-curves have three curvatures: Geodesic curvature (z), geodesic torsion (x), and normal curvature (y). For each of them, if avoided, a related curve type exists: geodesic curves, principle curvature lines and asymptotic curves. 3 Eike Schling, Denis Hitrec, Jonas Schikore and Rainer Barthel 3 METHOD 3.1 Minimal surface A minimal surface is the surface of minimal area between any given boundaries. Minimal surfaces have a constant mean curvature of zero. In nature such shapes result from an equilibrium of homogeneous tension, e.g. in a soap film. Various tools are capable of approximating minimal surfaces based on meshes, with varying degrees of precision and speed (Surface Evolver, Kangaroo-SoapFilm, Millipede, etc.). They are commonly based on a method by Pinkall and Polthier (1993) 8. The Rhino-plugin TeDa (Chair of Structural Analysis, TUM) provides a tool to model minimal surfaces as NURBS, based on isotropic pre-stress fields 9. Certain minimal surfaces can be modelled via their mathematical definition. This is especially helpful as a reference when testing the accuracy of other algorithms. 3.2 Asymptotic curves Geometrically, the local direction of an asymptotic curve can be found by intersecting the surface with its own tangent plane. We developed a custom VBScript for Grasshopper/Rhino to trace asymptotic curves on NURBS-surface using differential geometry. A detailed description of this method and the generation of accurate strip models is being published parallel at the Design Modelling Symposium 2017 1. Figure 5: Enneper surface with a) asymptotic curves b) principle curvature lines c) web of both networks d) strip model of the asymptotic network 3.3 Network design Asymptotic curve networks consist of two families of curves that follow a direction field. The designer can only pick a starting point, but cannot alter their path. If the surface is locally planar, the quadrilateral network forms a singularity with a higher valence. To achieve a homogeneous network, we take advantage of the bisecting property between asymptotic curves and principle curvature lines (Fig. 5) 10. By alternately drawing each curve and using their intersections as new starting point, we create an “isothermal” web with nearly quadratic cells 11. The node to node distance, measured along the asymptotic curves, is the only variable information needed to mark the intersections on the flat and straight strips before bending and twisting them into an asymptotic support structure. 4 Eike Schling, Denis Hitrec, Jonas Schikore and Rainer Barthel 4 DESIGN 4.1 General insights A minimal surface can be defined by one (a, b), two (c), or multiple (d, e) closed boundary-curves (Fig. 6). Symmetry properties can be used to create repetitive (a, b, c) or periodic (e) minimal surfaces. Boundary-curves may consist of straight lines (a), planar curves (d), or spatial curves (b, c, e). Straight or planar curves are likely to attract singularities (a, c, d). A well-integrated edge can also be achieved by modelling a larger surface and “cookie-cutting” the desired boundary.
Load more

Recommended publications
  • Abstract Book
    14th International Geometry Symposium 25-28 May 2016 ABSTRACT BOOK Pamukkale University Denizli - TURKEY 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 14th International Geometry Symposium ABSTRACT BOOK 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium Edited By: Dr. Şevket CİVELEK Dr. Cansel YORMAZ E-Published By: Pamukkale University Department of Mathematics Denizli, TURKEY All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means or whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder. Authors of papers in these proceedings are authorized to use their own material freely. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to: Assoc. Prof. Dr. Şevket CİVELEK Pamukkale University Department of Mathematics Denizli, TURKEY Email: [email protected] 2 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium May 25-28, 2016 Denizli, Turkey. Jointly Organized by Pamukkale University Department of Mathematics Denizli, Turkey 3 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 PREFACE This volume comprises the abstracts of contributed papers presented at the 14th International Geometry Symposium, 14IGS 2016 held on May 25-28, 2016, in Denizli, Turkey. 14IGS 2016 is jointly organized by Department of Mathematics, Pamukkale University, Denizli, Turkey. The sysposium is aimed to provide a platform for Geometry and its applications.
    [Show full text]
  • J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2019 This Course Is an Introduction to the Geometry of Smooth Curves and Surf
    J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2019 This course is an introduction to the geometry of smooth if the velocity never vanishes). Then the speed is a (smooth) curves and surfaces in Euclidean space Rn (in particular for positive function of t. (The cusped curve β above is not regular n = 2; 3). The local shape of a curve or surface is described at t = 0; the other examples given are regular.) in terms of its curvatures. Many of the big theorems in the DE The lengthR [ : Länge] of a smooth curve α is defined as subject – such as the Gauss–Bonnet theorem, a highlight at the j j len(α) = I α˙(t) dt. (For a closed curve, of course, we should end of the semester – deal with integrals of curvature. Some integrate from 0 to T instead of over the whole real line.) For of these integrals are topological constants, unchanged under any subinterval [a; b] ⊂ I, we see that deformation of the original curve or surface. Z b Z b We will usually describe particular curves and surfaces jα˙(t)j dt ≥ α˙(t) dt = α(b) − α(a) : locally via parametrizations, rather than, say, as level sets. a a Whereas in algebraic geometry, the unit circle is typically be described as the level set x2 + y2 = 1, we might instead This simply means that the length of any curve is at least the parametrize it as (cos t; sin t). straight-line distance between its endpoints. Of course, by Euclidean space [DE: euklidischer Raum] The length of an arbitrary curve can be defined (following n we mean the vector space R 3 x = (x1;:::; xn), equipped Jordan) as its total variation: with with the standard inner product or scalar product [DE: P Xn Skalarproduktp ] ha; bi = a · b := aibi and its associated norm len(α):= TV(α):= sup α(ti) − α(ti−1) : jaj := ha; ai.
    [Show full text]
  • AN INTRODUCTION to the CURVATURE of SURFACES by PHILIP ANTHONY BARILE a Thesis Submitted to the Graduate School-Camden Rutgers
    AN INTRODUCTION TO THE CURVATURE OF SURFACES By PHILIP ANTHONY BARILE A thesis submitted to the Graduate School-Camden Rutgers, The State University Of New Jersey in partial fulfillment of the requirements for the degree of Master of Science Graduate Program in Mathematics written under the direction of Haydee Herrera and approved by Camden, NJ January 2009 ABSTRACT OF THE THESIS An Introduction to the Curvature of Surfaces by PHILIP ANTHONY BARILE Thesis Director: Haydee Herrera Curvature is fundamental to the study of differential geometry. It describes different geometrical and topological properties of a surface in R3. Two types of curvature are discussed in this paper: intrinsic and extrinsic. Numerous examples are given which motivate definitions, properties and theorems concerning curvature. ii 1 1 Introduction For surfaces in R3, there are several different ways to measure curvature. Some curvature, like normal curvature, has the property such that it depends on how we embed the surface in R3. Normal curvature is extrinsic; that is, it could not be measured by being on the surface. On the other hand, another measurement of curvature, namely Gauss curvature, does not depend on how we embed the surface in R3. Gauss curvature is intrinsic; that is, it can be measured from on the surface. In order to engage in a discussion about curvature of surfaces, we must introduce some important concepts such as regular surfaces, the tangent plane, the first and second fundamental form, and the Gauss Map. Sections 2,3 and 4 introduce these preliminaries, however, their importance should not be understated as they lay the groundwork for more subtle and advanced topics in differential geometry.
    [Show full text]
  • J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2015 This Course Is an Introduction to the Geometry of Smooth Curves and Surf
    J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2015 This course is an introduction to the geometry of smooth The length of an arbitrary curve can be defined (Jordan) as curves and surfaces in Euclidean space (usually R3). The lo- total variation: cal shape of a curve or surface is described in terms of its Xn curvatures. Many of the big theorems in the subject – such len(α) = TV(α) = sup α(ti) − α(ti−1) . as the Gauss–Bonnet theorem, a highlight at the end of the ··· ∈ t0< <tn I i=1 semester – deal with integrals of curvature. Some of these in- tegrals are topological constants, unchanged under deforma- This is the supremal length of inscribed polygons. (One can tion of the original curve or surface. show this length is finite over finite intervals if and only if α Usually not as level sets (like x2 + y2 = 1) as in algebraic has a Lipschitz reparametrization (e.g., by arclength). Lips- geometry, but parametrized (like (cos t, sin t)). chitz curves have velocity defined a.e., and our integral for- Of course, by Euclidean space [DE: euklidischer Raum] we mulas for length work fine.) n mean the vector space R 3 x = (x1,..., xn) with the stan- If J is another interval and ϕ: J → I is an orientation- dard scalar product [DE: Skalarprodukt] (also called an in- preserving homeomorphism, i.e., a strictly increasing surjec- P n ner product√ ) ha, bi = a · b = aibi and its associated norm tion, then α◦ϕ: J → R is a parametrized curve with the same |a| = ha, ai).
    [Show full text]
  • A Note on Inextensible Flows of Curves on Oriented Surface
    A Note on Inextensible Flows of Curves on Oriented Surface Onder Gokmen Yildiza, Soley Ersoy b, Melek Masal c a Department of Mathematics, Faculty of Arts and Sciences Bilecik University, Bilecik/TURKEY b Department of Mathematics, Faculty of Arts and Sciences Sakarya University, Sakarya/TURKEY c Department of Mathematics Teaching , Faculty of Education Sakarya University, Sakarya/TURKEY Abstract In this paper, the general formulation for inextensible flows of curves on oriented surface in R3 is investigated. The necessary and sufficient con- ditions for inextensible curve flow lying an oriented surface are expressed as a partial differential equation involving the geodesic curvature and the geodesic torsion. Moreover, some special cases of inextensible curves on oriented surface are given. Mathematics Subject Classification (2010): 53C44, 53A04, 53A05, 53A35. Keywords: Curvature flows, inextensible, oriented surface. 1 Introduction It is well known that many nonlinear phenomena in physics, chemistry and biology are described by dynamics of shapes, such as curves and surfaces. The evolution of curve and surface has significant applications in computer vision arXiv:1106.2012v1 [math.DG] 10 Jun 2011 and image processing. The time evolution of a curve or surface generated by its corresponding flow in R3 -for this reason we shall also refer to curve and surface evolutions as flows throughout this article- is said to be inextensible if, in the former case, its arclength is preserved, and in the latter case, if its intrinsic curvature is preserved. Physically, the inextensible curve flows give rise to motions in which no strain energy is induced. The swinging motion of a cord of fixed length, for example, or of a piece of paper carried by the wind, can be described by inextensible curve and surface flows.
    [Show full text]
  • International Geometry Symposium Abstracts Book
    16th International Geometry Symposium July 4-7, 2018 Manisa Celal Bayar University, Manisa-TURKEY 16TH INTERNATIONAL GEOMETRY SYMPOSIUM ABSTRACTS BOOK 1 16th International Geometry Symposium July 4-7, 2018 Manisa Celal Bayar University, Manisa-TURKEY Proceedings of the 16th International Geometry Symposium Edited By: Prof. Dr. Mustafa KAZAZ Res. Assist. Dr. Burak ŞAHİNER Res. Assist. Onur KAYA E-published by: Manisa Celal Bayar University All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means or whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder. Authors of papers in these proceedings are authorized to use their own material freely. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to: Prof. Dr. Mustafa KAZAZ Manisa Celal Bayar University [email protected] 2 16th International Geometry Symposium July 4-7, 2018 Manisa Celal Bayar University, Manisa-TURKEY Proceedings of the 16th International Geometry Symposium July 4-7, 2018 Manisa, Turkey Jointly Organized By Manisa Celal Bayar University 3 16th International Geometry Symposium July 4-7, 2018 Manisa Celal Bayar University, Manisa-TURKEY FOREWORD Hosted by Manisa Celal Bayar University between July 4-7, 2018, the 16th International Geometry Symposium was held in Manisa, a city of learning throughout history. Undergraduate students aiming to do scholarly studies as well as new researchers had a great opportunity of getting together with highly experienced researchers. In light of scientific developments in Geometry, presentations were made, and discussions were held, thus paving the way for new research.
    [Show full text]
  • Surfaces Family with Common Smarandache Asymptotic Curve
    Bol. Soc. Paran. Mat. (3s.) v. 34 1 (2016): 9–20. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm Surfaces Family With Common Smarandache Asymptotic Curve Gülnur Şaffak Atalay and Emin Kasap abstract: In this paper, we analyzed the problem of consructing a family of surfaces from a given some special Smarandache curves in Euclidean 3-space. Using the Frenet frame of the curve in Euclidean 3-space, we express the family of surfaces as a linear combination of the components of this frame, and derive the necessary and sufficient conditions for coefficents to satisfy both the asymptotic and isopara- metric requirements. Finally, examples are given to show the family of surfaces with common Smarandache curve. Contents 1 Introduction 9 2 Preliminaries 10 3 Surfaces with common Smarandache asymptotic curve 11 4 Examples of generating simple surfaces with common Smarandache asymptotic curve 17 1. Introduction In differential geometry, there are many important consequences and properties of curves [1], [2],[3]. Researches follow labours about the curves. In the light of the existing studies, authors always introduce new curves. Special Smarandache curves are one of them. Special Smarandache curves have been investigated by some dif- ferential geometers [4,5,6,7,8,9,10]. This curve is defined as, a regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is Smarandache curve [4]. A.T. Ali has introduced some special Smarandache curves in the Euclidean space [5]. Special Smarandache curves according to Bishop Frame in Euclidean 3-space have been investigated by Çetin et al [6].
    [Show full text]
  • DIFFERENTIAL GEOMETRY: a First Course in Curves and Surfaces
    DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces Preliminary Version Fall, 2015 Theodore Shifrin University of Georgia Dedicated to the memory of Shiing-Shen Chern, my adviser and friend c 2015 Theodore Shifrin No portion of this work may be reproduced in any form without written permission of the author, other than duplication at nominal cost for those readers or students desiring a hard copy. CONTENTS 1. CURVES.................... 1 1. Examples, Arclength Parametrization 1 2. Local Theory: Frenet Frame 10 3. SomeGlobalResults 23 2. SURFACES: LOCAL THEORY . 35 1. Parametrized Surfaces and the First Fundamental Form 35 2. The Gauss Map and the Second Fundamental Form 44 3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory 57 4. Covariant Differentiation, Parallel Translation, and Geodesics 66 3. SURFACES: FURTHER TOPICS . 79 1. Holonomy and the Gauss-Bonnet Theorem 79 2. An Introduction to Hyperbolic Geometry 91 3. Surface Theory with Differential Forms 101 4. Calculus of Variations and Surfaces of Constant Mean Curvature 107 Appendix. REVIEW OF LINEAR ALGEBRA AND CALCULUS . 114 1. Linear Algebra Review 114 2. Calculus Review 116 3. Differential Equations 118 SOLUTIONS TO SELECTED EXERCISES . 121 INDEX ................... 124 Problems to which answers or hints are given at the back of the book are marked with an asterisk (*). Fundamental exercises that are particularly important (and to which reference is made later) are marked with a sharp (]). August, 2015 CHAPTER 1 Curves 1. Examples, Arclength Parametrization We say a vector function f .a; b/ R3 is Ck (k 0; 1; 2; : : :) if f and its first k derivatives, f , f ,..., W ! D 0 00 f.k/, exist and are all continuous.
    [Show full text]
  • Asymptotic Path Curves
    Asymptotic Path Curves N. C. Thomas, C.Eng, MIET Asymptotic Curves The results described below were first published more concisely in Ref. 2. A curve C in a surface S passing through a point P in S possesses curvature at that point. In the tangent plane to S at P a flat pencil of tangents to the surface exists, and a curve in the surface passing though P will have one of the lines of the pencil as its tangent at P. The curvature of all curves at P touching the set of tangents at P may be positive, zero or negative depending upon the nature of the surface. If a surface has points where the curvature of C may be positive or negative for different tangents then it is said to have negative Gaussian curvature. There will be two tangents where C has zero curvature, and the directions of those tangents are called asymptotic directions. An asymptotic line (or curve) in a surface is such that all its tangents are asymptotic directions. Generally two asymptotic curves pass through every point of a surface which has negative Gaussian curvature. A ruled hyperboloid is an example of a surface with negative curvature, as will be explained below, while a sphere or ellipsoid has positive curvature and therefore no asymptotic curves. Thus an asymptotic curve is a kind of boundary between positive and negative curvature on a surface. For example, consider a ruled hyperboloid: Figure 1 The red plane intersects it in a circle, a curve which has positive curvature, while the blue plane intersects it in a hyperbola, which has negative curvature.
    [Show full text]
  • Chapter 6 Some Special Curves on Surfaces
    Chapter 6 Some Special Curves on Surfaces In this chapter we introduce three families of curves on surfaces, namely, principal, asymptotic and geodesic curves. Principal curves follow paths of maximal bending, asymptotic curves follow paths of zero bending and geodesicscurvesalwaysgo"striaghtahead"inthesurface. 6.1 Vector Fields Along Curves Our discussion begins with an introduction to vector fields along curves. Definition 135 Let α :(a, b) R3 be a curve given componentwise by α(t)= (x(t),y(t),z(t)).Avectorfield→V along α is a function defined on (a, b) such that V(t) R3 for all t (a, b). Each vector field V along α has the form ∈ α(t) ∈ V(t)=(V1(t),V2(t),V3(t))α(t) , where Vi :(a, b) R. Avectorfield V along α is differentiable if each Vi is differentiable; the→ derivative of a differentiable vector field V along α is the vector field V0 along α defined by V0(t)=(V10(t),V20(t),V30(t))α(t) . The velocity vector field along α is defined by α0(t)=(x0(t),y0(t),z0(t))α(t) ; the acceleration vector field along α is defined by 93 94 CHAPTER 6. SOME SPECIAL CURVES ON SURFACES α00(t)=(x00(t),y00(t),z00(t))α(t) . All vector fields henceforth are assumed to be infinitely differentiable, i.e., smooth. Definition 136 Let α :(a, b) R3 be a curve, let f :(a, b) R be a differentiable function and let V→, W be vector fields along α. For al→ l t (a, b), define ∈ a.
    [Show full text]
  • DIFFERENTIAL GEOMETRY: a First Course in Curves and Surfaces
    DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces Preliminary Version Summer, 2016 Theodore Shifrin University of Georgia Dedicated to the memory of Shiing-Shen Chern, my adviser and friend c 2016 Theodore Shifrin No portion of this work may be reproduced in any form without written permission of the author, other than duplication at nominal cost for those readers or students desiring a hard copy. CONTENTS 1. CURVES.................... 1 1. Examples, Arclength Parametrization 1 2. Local Theory: Frenet Frame 10 3. SomeGlobalResults 23 2. SURFACES: LOCAL THEORY . 35 1. Parametrized Surfaces and the First Fundamental Form 35 2. The Gauss Map and the Second Fundamental Form 44 3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory 57 4. Covariant Differentiation, Parallel Translation, and Geodesics 66 3. SURFACES: FURTHER TOPICS . 79 1. Holonomy and the Gauss-Bonnet Theorem 79 2. An Introduction to Hyperbolic Geometry 91 3. Surface Theory with Differential Forms 101 4. Calculus of Variations and Surfaces of Constant Mean Curvature 107 Appendix. REVIEW OF LINEAR ALGEBRA AND CALCULUS . 114 1. Linear Algebra Review 114 2. Calculus Review 116 3. Differential Equations 118 SOLUTIONS TO SELECTED EXERCISES . 121 INDEX ................... 124 Problems to which answers or hints are given at the back of the book are marked with an asterisk (*). Fundamental exercises that are particularly important (and to which reference is made later) are marked with a sharp (]). June, 2016 CHAPTER 1 Curves 1. Examples, Arclength Parametrization 3 Ck We say a vector function fW .a; b/ ! R is (k D 0; 1; 2; : : :) if f and its first k derivatives, f0, f00,..., f.k/, exist and are all continuous.
    [Show full text]
  • Differential Geometry Part V: Shape Operators
    Differential Geometry Part V: Shape Operators Jay Havaldar We can now study the shape of surfaces through the lens of linear algebra, namely a symmetric operator called the shape operator. We draw comparison with the Frenet formulas and their generalizations using the connection forms1, which describe curves in terms of their curvature and torsion, to develop similar machinery describing surfaces. Finally, we generalize the unique- ness theorem for curves2 to a congruence theorem for surfaces, although that will have to wait until Part VI. 0.1 The Shape Operator of M ⊂ R3 Recall how we defined the covariant derivative of a vector field Z with respect to a vector v, denoted as rvZ. By an earlier lemma, we were able to show that in Euclidean space: d rvZ = Z(p + tv) jt= 0 Xdt = v[wi]Ui This definition still more or less makes sense for any vector field on a surface if we pick v tangent to a surface M so that the derivative is defined. In addition to the earlier method of computation, we can also refer to the definition of derivatives on surfaces to obtain: r Z = (Z )0(0) v Xα = v[wi]Ui 0 Where Zα is the vector field Z restricted to a curve α through a point p where α (0) = v. Using the covariant derivative, we now define the shape operator. Note that if a surface is orientable, it has a (differentiable) unit normal vector field U everywhere. If it is connected, there are exactly two (in opposite directions). For a non-orientable surface, we still have local parameter curves, which allow us to create a local unit normal vector field.
    [Show full text]